Contrary to the case of classical mechanics, QM relies on a non-commutative algebraic structureFor details on the mathematical structure of QM and its physical content the reader may refer e.g. to [1]}, and [2]}. In this essay, it is assumed that the reader has some familiarity with the standard formalism of QM.: quantum observables generally do not commute, meaning that for any pair of operators \(A,B\)  we have \([A,B]=AB-BA\ne 0\) . From a physical perspective this fact entails that if one performs a measurement of \(A\)  on a quantum system followed by a measurement of \(B\) , in general one will obtain different results inverting the order of these observations. Moreover, in virtue of the non-commutative algebra of quantum theory, measuring an operator \(C=A+B\)  is not typically equivalent to measuring \(A\)  and \(B\)  independently and then adding the respective results, since \(A,B\)  will be in general incompatible (cf. [3]}, p. 77). This feature of QM constitutes the formal basis to prove the Heisenberg uncertainty relation, a theorem of quantum theory which reflects the operational inability to simultaneously prepare/measure the values of incompatible operators with arbitrary precision. Such a result, in turn, is usually interpreted ontologically in the sense that quantum systems do not instantiate definite properties in non-measurement situations (cf. [1]}).
